Problem: The lifespans of turtles in a particular zoo are normally distributed. The average turtle lives $120$ years; the standard deviation is $4.8$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a turtle living between $129.6$ and $134.4$ years.
Explanation: $120$ $115.2$ $124.8$ $110.4$ $129.6$ $105.6$ $134.4$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $120$ years. We know the standard deviation is $4.8$ years, so one standard deviation below the mean is $115.2$ years and one standard deviation above the mean is $124.8$ years. Two standard deviations below the mean is $110.4$ years and two standard deviations above the mean is $129.6$ years. Three standard deviations below the mean is $105.6$ years and three standard deviations above the mean is $134.4$ years. We are interested in the probability of a turtle living between $129.6$ and $134.4$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the turtles will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the turtles will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of turtles between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular turtle living between $129.6$ and $134.4$ years is $\color{orange}{2.35\%}$.